Kunze Linear Algebra · Problem 2.3.9

The key is to test linear independence directly: assume a linear combination of these three new vectors equals zero, then use the original independence of $\alpha$, $\beta$, $\gamma$ to force all coefficients to be zero. When you expand the combination and collect terms by the original vectors, you'll get a system of equations in the original coefficients—the constraint that $\alpha$, $\beta$, $\gamma$ are independent means each original vector's coefficient must vanish separately, which cascades back to your original unknowns.

The key is to test linear independence directly: assume a linear combination of these three new vectors equals zero, then use the original independence of $\alpha$, $\beta$, $\gamma$ to force all coefficients to be zero. When you expand the combination and collect terms by the original vectors, you'll get a system of equations in the original coefficients—the constraint that $\alpha$, $\beta$, $\gamma$ are independent means each original vector's coefficient must vanish separately, which cascades back to your original unknowns.